Related papers: Adjoint and duality for rank-metric codes in a ske…
We investigate two classes of extended codes and provide necessary and sufficient conditions for these codes to be non-GRS MDS codes. We also determine the parity check matrices for these codes. Using the connection of MDS codes with arcs…
It is well-known that the dimension of optimal anticodes in the rank-metric is divisible by the maximum m between the number of rows and columns of the matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes are the…
Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [Kurihara et al. 2012] and in the CSS construction of quantum codes [Ketkar et al. 2006]. The important parameters are (1) the codimension, (2) the…
MDS codes have diverse practical applications in communication systems, data storage, and quantum codes due to their algebraic properties and optimal error-correcting capability. In this paper, we focus on a class of linear codes and…
Sum-rank metric codes are a natural extension of both linear block codes and rank-metric codes. They have several applications in information theory, including multishot network coding and distributed storage systems. The aim of this…
In this paper we introduce and investigate rank-metric intersecting codes, a new class of linear codes in the rank-metric context, inspired by the well-studied notion of intersecting codes in the Hamming metric. A rank-metric code is said…
The applications of additive codes mainly lie in quantum error correction and quantum computing. Due to their applications in quantum codes, additive codes have grown in importance. In addition to this, additive codes allow the…
We construct a family of (n,k) convolutional codes with degree \delta in {k,n-k} that have a maximum distance profile. The field size required for our construction is of the order n^{2\delta}, which improves upon the known constructions of…
We describe families of nonassociative finite unital rings that occur as quotients of natural nonassociative orders in generalized nonassociative cyclic division algebras over number fields. These natural orders have already been used to…
In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring. It has been shown in [3] that only certain combinations of the…
We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in the…
In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and…
Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the exceptional ones, as defined in 2018 by the first author and Zhou, for which…
In this paper, we study properties of rank metric codes in general and maximum rank distance (MRD) codes in particular. For codes with the rank metric, we first establish Gilbert and sphere-packing bounds, and then obtain the asymptotic…
Rank-metric codes are subspaces of matrices over finite fields endowed with the rank metric and admit a natural tensorial representation. The tensor rank provides a measure of the minimal size of a decomposition of a code into rank-one…
Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several…
The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming…
The weight distribution of an error correcting code is a crucial statistic in determining it's performance. One key tool for relating the weight of a code to that of it's dual is the MacWilliams Identity, first developed for the Hamming…
Maximum distance separable (MDS) and near maximum distance separable (NMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes due to their algebraic properties and excellent…
We introduce a new technique to construct rank-metric codes using the arithmetic theory of Drinfeld modules over global fields, and Dirichlet Theorem on polynomial arithmetic progressions. Using our methods, we obtain a new infinite family…